Issue 
EPJ Appl. Metamat.
Volume 10, 2023



Article Number  5  
Number of page(s)  15  
DOI  https://doi.org/10.1051/epjam/2023002  
Published online  12 July 2023 
https://doi.org/10.1051/epjam/2023002
Research Article
Reflection and transmission of nanoresonators including biisotropic and metamaterial layers: opportunities to control and amplify chiral and nonreciprocal effects for nanophotonics applications
Gomel State Technical University, October ave. 48, Gomel 246746, Belarus
^{*} email: starodub@tut.by
Received:
6
December
2022
Accepted:
10
May
2023
Published online: 12 July 2023
Electromagnetic waves reflected from and transmitted through the multilayer nanoresonators including the main layer made of a biisotropic material or metamaterial sandwiched between dielectric, epsilonnearzero or metallic spacer layers have been analytically modeled. The numerical and graphical analysis, based on the exact solution of the electromagnetic boundary problem, confirms opportunities to use such nanoresonators as utracompact polarization converters. The proposed systems are characterized by wide ranges of parameters and significantly reduced (subwavelength) thicknesses. The spacer layers can provide modification, control, and amplification of chiral and nonreciprocal effects for the reflected and transmitted radiation. The concept can be realized for various geometries of dielectric, epsilonnearzero, metallic, biisotropic, metamaterial layers and used to develop new ultrathin, large area, and relatively easytomanufacture polarization and other devices for nanophotonics.
Key words: Nanoresonators / ultrathin polarization converters / biisotropic media / epsilon nearzero metamaterials / chiral nihility metamaterials
© E. Starodubtsev, Published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The present paper is the second of two ones (the first is [1] where the detailed problem statement and general analytical model were discussed) devoted to the modeling of optical properties of the planelayered nanoresonators including the main layer made of a biisotropic material or metamaterial (MM) sandwiched between pairs of dielectric, epsilonnearzero (ENZ) or metallic spacer layers. Such onedimensional nanometric systems (that are relatively easy to realize using modern thin film and nanocomposite technologies) demonstrate the optical properties that can be of interest for many nanophotonics applications. The main goal of both works is to investigate chiral and nonreciprocal effects in the systems of subwavelength thicknesses, possibilities to control and “amplify'' these effects by varying geometry and materials of the nanoresonator components, exciting radiation characteristics. There is a large number of both classical (e.g. [2–5]) and recent (see, e.g. [6–10] and numerous references in these papers) works in the field of chiral natural (gyrotropic) and artificial media. However, new opportunities, arising from the rapid development of electrodynamics and technology of electromagnetic MMs [1,11–21], require further investigations and these opportunities are seemed to be very perspective for various applications of such media.
This paper uses the analytical model proposed in [1] and based on the accurate solution of the corresponding boundary electromagnetic problem for monochromatic waves, to investigate the features of reflected and transmitted radiation (the main properties of the proper waves inside of the system were considered in [1]). The present paper includes the detailed numerical and graphical analysis of the energy and polarization characteristics of the waves outside of the nanoresonator pointing to the new possibilities for applications of such systems as ultrathin polarization and phase converters, filters, modulators, metasurfaces and other components of nanophotonics devices.
Let us note that the investigated in the work reflection and transmission characteristics of the nanoresonators with absorbing biisotropic and MM layers of subwavelength thicknesses, apparently, have not yet been considered in detail (in contrast to the investigations of classical “macroscopic” chiral and nonreciprocal layered systems, e.g. [3–5]). Primarily, we consider the new features of the biisotropic effects in reflection and transmission (for both intensity and polarization characteristics) of such multilayers due to the “addition” of different (dielectric, ENZ, metallic, MM) components to the ultrathin layered systems.
The work has the following structure. For the sake of completeness, the brief problem statement, the analytical model description and main results from [1] (used for the numerical analysis in the present work) are given in Section 2. Main parameters and dependences for the numerical analysis are considered in Section 3. The polarization and intensity characteristics of the waves reflected from and transmitted through the nanoresonator are investigated in Sections 4 and 5, respectively. Special attention is paid to the analysis of the polarization effects of birefringence, dichroism, nonreciprocity and clarification of the conditions when it is possible to control such effects and “optimize” their characteristics by changing parameters of the nanoresonator and exciting radiation. In doing so, the cases of both “conventional” and MM components of the system are considered and compared. Section 6 includes the discussion of the obtained results and summarizes the paper.
2 Statement of the problem and general relations
We consider the interaction of an electromagnetic plane monochromatic wave with a layered system under oblique incidence [1]. A geometry of the multilayer and the corresponding electromagnetic boundary problem is illustrated in Figure 1. The layers have thicknesses d_{1,2,3} (further lower indexes, l = 0, 1, 2, 3, 4 relate to the quantities characterizing the materials of the superstrate, multilayer components, and substrate, respectively). The superstrate and substrate media are assumed to be isotropic and semiinfinite (optically thick). So, there is only a transmitted wave inside of the substrate. Coordinate axis Z is perpendicular to the boundaries of the layers, plane XZ is the incidence plane. The relative intensities of the incident, reflected, and transmitted waves, I_{in}, I_{r}, I_{t}, and polarization characteristics of these waves are considered at boundaries z = 0 and z = d_{1} + d_{2} + d_{3} (Fig. 1).
The materials of media 0, 1, 3, 4 are supposed to be isotropic, they are characterized by scalar dielectric permittivities ε_{0,1,3,4} and magnetic permeabilities μ_{0,1,3,4}. We assume that media 0, 4 (1, 2, 3) are nonabsorbing (absorbing), correspondingly. Thus, quantities ε_{0,4}, μ_{0,4} are real and ε_{1,2,3}, μ_{1,2,3} are complex in the general case. According to the phase multiplier choice for the fields in the system [1] as exp [i (k_{1}x + k_{3}z) − iωt] (k = (k_{1}, 0, k_{3}) is a wave vector) and the causality conditions, the imaginary parts of permittivities and permeabilities for all the multilayer components are positive. The real parts of these quantities can be both positive and negative, so, we also consider the cases when the media are MMs. Denotations
a ' = Re (a), a '' = Im (a)
are used as well for the real and imaginary parts of scalar or vector quantities.
For the homogeneous incident, reflected and transmitted waves in the nonabsorbing superstrate and substrate media, the real angles of incidence (reflection), ψ_{in}, and refraction, ψ_{t}, are used (Fig. 1). These waves are assumed to be elliptically polarized in the general case. Orientations of the major axes of polarization ellipses (for the corresponding vectors of electric field strengths) in the related phase planes X'Y, X''Y, X'''Y are determined by angles θ_{in}, θ_{r}, θ_{t}, respectively (see the inserts in Fig. 1).
The main layer material is biisotropic and it is described by the constitutive relations including cross magnetodielectric terms [2–5]: D = εE + (χ + iα) H, B = μH + (χ − iα) E, where scalar (complex, in general) quantities ε = ε_{2}, μ = μ_{2}, α, χ (Fig. 1) are dielectric permittivity, magnetic permeability, chirality and nonreciprocity parameters, respectively. In the particular cases α ≠ 0, χ = 0 and α = 0, χ ≠ 0, these constitutive relations correspond to chiral (or naturally gyrotropic) and Tellegen's media [2–5].
To investigate numerically energy and polarization characteristics of the reflected and transmitted waves (Fig. 1), we use the previous analytical results [1]. The relative intensities of the reflected (at the boundary z = 0) and transmitted (at z = d_{1} + d_{2} + d_{3}) radiation are determined by the expressions
where complex scalars , , , characterize the fields in the media ν = 0, 4 (Fig. 1) according to the relations
In equations (2), the general for all the waves phase factor exp [ik_{0} (m_{1}x ± η_{v}z) − iωt] is omitted, (choosing ), m = (m_{1}, 0, m_{3}) = k/k_{0} is a complex normalized wave vector or so called refraction vector [2] (k_{0} is the wave number for vacuum), the upper indexes “+” and “− ” correspond to the quantities describing transmitted (ν = 4) and reflected (ν = 0) proper waves.
Quantities , for the reflected wave in the superstrate and , for the transmitted wave in the substrate are determined from the boundary problem solution using the given parameters , , m_{1} characterizing the incident wave [1]:
where , , c is the speed of light in vacuum, τ_{in} and I_{in} are ellipticity and intensity of the incident wave (τ_{in} = 0 for linear and τ_{in} = +1 for right and τ_{in} = − 1 for left circular polarizations), θ_{in} is an angle between the incidence plane (XZ) and the major axis of polarization ellipse lying in the phase plane of the incident wave, ψ_{in} is the incidence angle (Fig. 1).
The polarization parameters of the reflected and transmitted waves are determined by the relations of the components of these waves in the corresponding phase planes X''Y and X'''Y (Fig. 1) [1]:
where ψ_{in} and ψ_{t} are the incidence and refraction angles (Fig. 1), . Using the approach [2], parameters ξ_{r,t} determine ellipticities τ_{r,t} and polarization azimuths θ_{r,t} (Fig. 1) of the reflected and transmitted waves according to the expressions:
where ξ = ξ_{r,t}, θ = θ_{r,t}. In these expressions, we choose τ > 0 if ξ '' > 0 (right polarization), τ < 0 if ξ '' < 0 (left polarization), τ = 0 (ξ '' = 0) for linear polarization. The detailed description of the analytical model used here for the numerical and graphical analysis was given in [1].
The used analytical model is based on the assumptions considered in detail in [1]. In particular, the frequency dispersion of the electromagnetic parameters characterizing layer materials is not taken into account explicitly in the work. Instead, we consider below some “typical” values of parameters of different material layers that are characteristic for wide frequency ranges and can be realized using modern nanocomposite technologies. Such approach is caused by the main goal of the numerical analysis below, that is to confirm and illustrate (for specific and typical data) the opportunities to control and amplify the biisotropic effects for the reflection and transmission of the considered nanoresonator systems. Furthermore, we use a “purely phenomenological” description of rather complicated (chiral, nonreciprocal, MM, ENZ) electromagnetic media with effective material parameters without specifying methods of their realization. The consideration of these methods and their features (e.g. account of spatial dispersion of the layered materials that can demonstrate properties of effective ENZ media [13,22–24]) goes beyond the present paper. However, these electromagnetic materials have been investigated for a long time, and various technologies to obtain and homogenize such materials (including also local analytical models for ENZ MMs) are sufficiently considered and experimentally confirmed (e.g., see [6–11,14,22,23] and numerous references in these works).
Fig. 1 The multilayer scheme and geometry of the corresponding electromagnetic boundary problem. The superstrate and substrate media are isotropic and semiinfinite. The multilayer includes spacer layers 1, 3 with thicknesses d_{1,3}, whose electromagnetic properties correspond to isotropic media (in particular, ENZ MMs), and the main biisotropic layer 2 with thickness d_{2}. Relative intensities I_{in}, I_{r}, I_{t}, ellipticities τ_{in}, τ_{r}, τ_{t}, polarization azimuths θ_{in}, θ_{r}, θ_{t}, and wave vectors k_{in}, k_{r}, k_{t} correspond to the incident, reflected, and transmitted waves, ψ_{in} and ψ_{t} are the angles of incidence and refraction, respectively. The fields inside of the multilayer are not shown. The inserts bounded by dotted lines illustrate orientations of the major axes of polarization ellipses of the incident, reflected and transmitted waves in the corresponding phase planes X'Y, X''Y, X'''Y, respectively. Additional notation is given in the main text. 
3 Main parameters and dependences for numerical analysis
The numerical analysis aims to investigate the characteristic properties of the reflected and transmitted radiation for subwavelength thicknesses of the layers or whole multilayer. At the same time, particular attention is paid to search the regimes of effective polarization conversion of the radiation. The detailed description of the analytical model and parameters (that were used for the numerical analysis below) were given in Section 2 of the previous paper [1]. In particular, the following complex values of parameters are used by default for the main layer material: dielectric permittivity, ε_{2} = 3 + 10^{−2}i, magnetic permeability, μ_{2} = 1.5 + 10^{−2}i, chirality (α) and nonreciprocity (χ) parameters, α = (5/4) χ = 5·10^{−3} (1 + i). Let us note that the choice of rather large α, χ values (that do not contradict the known restrictions on the α '', χ '' values for absorbing media [3,4]) is due to the requirement of clear visualization of the considered effects in the graphs below. However, the analyzed further effects also take place for smaller α, χ values. As the materials of the spacer layers, we use typical dielectric, ENZ, and metallic materials. We consider three types of the same materials of layers 1, 3 that will be called for brevity as: (i) “dielectric” layers, with parameters ε_{1,3} = 4 +0.01i, d_{1,3} = 0.15 μm; (ii) “ENZ” layers, ε_{1,3} = 0.01 + 0.01i, d_{1,3} = 0.1 μm; (iii) “metallic” layers, ε_{1,3} = −13 + 0.5i, d_{1,3} = 0.01 μm. For the numerical analysis, these materials are assumed to be nonmagnetic (μ_{1,3} = 1). For all the cases, subwavelength thicknesses d_{1} = d_{3} are chosen for a given d_{2} value and a material type to obtain the maximal effects of the spacer layers on the dependences under investigation. Also, we use the value λ = 0.65 μm for the incident wave wavelength. For all the figures, air is assumed to be the superstrate and substrate media (ε_{0,4} = μ_{0,4} = 1), and the thicknesses of layers 1, 3 are equal (d_{1} = d_{3}). The considered below effects also take place for the cases ε_{0} ≠ ε_{4}, μ_{0} ≠ μ_{4}, d_{1} ≠ d_{3}, and when changing quantities ε_{2}, μ_{2}, α, χ, d_{1,2,3}, λ in wide ranges. When the properties of the layers are different from the ones described above, the related values of parameters are given in captions to the figures below.
As basic quantities for the analysis, we consider I_{r,t}, τ_{r,t}, θ_{r,t} that are relative intensities (intensities normalized to the incident wave intensity), ellipticities, and polarization azimuths, respectively, of the reflected and transmitted waves (to which low indexes r and t correspond), Figure 1. First of all, we investigate the dependences of I_{r,t}, τ_{r,t}, θ_{r,t} on electromagnetic properties of the materials of layers 1, 2, 3, multilayer geometry (that is determined by thicknesses d_{1,2,3}), incidence angle (ψ_{in}) and polarization of the incident wave. All the angles in the graphs below (ψ_{in}, θ_{r,t}) are calculated in the relative units of radians/π.
The data in Figure 2 illustrate characteristics of the reflected from and transmitted through the nanoresonator waves under the variation of dielectric properties of the spacer layers and the incident wave polarization (linear TM and TE polarizations are equivalent for the normal incidence, panels a, b). These data correspond to the properties of the field inside of the biisotropic layer considered in Figure 6 of [1]. The changes of can significantly affect the quantities in the Figure 2 graphs. In all Figure 2 graphs, solid lines describe relative intensities I_{r,t}. The small values of the absorption parameters and thicknesses of the layers lead to the correspondence of the parameters ranges where low reflection and high transmission take place simultaneously.
The graphs for the polarization parameters of the reflected and transmitted waves (τ_{r,t}, θ_{r,t}) in all the panels of Figure 2, besides (b,c,d), can have resonancelike features for the definite ranges of values. The numerical analysis shows that these features (which determine, in particular, the crosspolarization effects for the reflected and transmitted waves [3,16,25]) are due to several reasons. The τ_{r}, θ_{r} peaks in panel (a) result from the nonreciprocity of the main layer material (χ ≠ 0). The peaks of τ_{r,t}, θ_{r,t} in panels (e,f) are mainly determined by the chirality parameter (α ≠ 0). The resonancelike features in cases (g,h,l,m) take place also when α = χ = 0, so they are mainly caused by the complex mode structure of the field inside of the nanoresonator under the oblique incidence of the circularly polarized incident wave (see alsoFigs. 6d, f in [1]). The abrupt changes of quantities τ_{r,t}, θ_{r,t} correspond, as a rule, to small values of I_{r} (I_{r} ≈ 0.01 ÷ 0.04, panels (a,e, g,l)) and rather large values of I_{t} (I_{t} ≈ 0.8 ÷ 1 at , panels (f,h,m)).
When changing the sign for the case in panel (b), angle θ_{t} also changes its sign, that is, the polarization plane of the transmitted wave rotates by the same angle to the opposite side. According to the used method to determine polarization characteristics (see the choice of angles θ_{r,t} in Fig. 1 and relations (21) of [1]), values θ_{r,t} < 0 (θ_{r,t} > 0) correspond to the right (left) rotation of the major axis of polarization ellipses (or to the polarization plane rotations for linearly polarized waves).
In going from the above considered data to the MM cases (the graphical data are not shown), the following basic effects take place. When (, that corresponds to the proper wave parameters in Figure 7 of [1]), we have the condition I_{t} ≈ 0 for all the data in Figure 2. Herewith, for the cases in panels (g,l), the abrupt changes of τ_{r} from 1 down to −1 (and vice versa) take place under the conditions , I_{r} ≈ 0.8 ÷ 0.9. For the other MM case (, , that corresponds to the parameters of proper waves in Fig. 8 of [1]), differences of the peak structure of τ_{r}, θ_{r} are exhibited for cases (a,e). In this case, the other graphs are weakly changed in comparison with the ones in Figure 2.
Fig. 2 The effect of the real part of the spacer layers permittivity on the parameters of reflected (I_{r}, τ_{r}, θ_{r}) and transmitted (I_{t}, τ_{t}, θ_{t}) waves ( is used as the abscissa axis for all the graphs). The following values of parameters are used: d_{1,3} = 0.15 μm, d_{2} = 0.25 μm, , ε_{2} = 3 +0.01i, μ_{2} = 1.5 + 0.01i. The values of ψ_{in} and the incident wave polarization (RCP and LCP for right and left circular polarizations, here and below) are given above the corresponding graph panels. The other values of parameters correspond to the ones given in the main text (here and in the figures below). 
4 Characteristic properties of reflected radiation
The dependences of the reflected wave characteristics on various parameters of the multilayer (materials and geometry) and the incident radiation (mainly, polarization and incidence angle) are given in Figures 3–6. The data in Figure 3 illustrate the dependences of the reflected wave relative intensity on the incidence angle for various polarizations of the exciting wave and properties of layers 1, 3.
It is seen that the spacer layers can lead to the significant changes in the dependences. The approximate analog of Brewster's effect can take place for all the considered polarizations (panels (a–c), d, (g–i) in Fig. 3). In these cases, the minimal I_{r} values are of the order of 10^{−4} ÷ 10^{−2}. With that, the spacer layers (especially, for the case d_{1,3} ≠ 0) can enable both the realization of the effect (panels (b,c,h,i)) and the pronounced shifts of Brewster's angle values (a,d,g).
Accounting for MM properties of the main layer material, the numerical analysis shows that the minimal differences from Figure 3 data take place when . In particular, in this case, the graphs are almost the same as in Figure 3 under the condition d_{1,3} ≠ 0. For , , the approximate Brewster effect is not practically realized, though the pronounced minima of dependences I_{r} (ψ_{in}) can take place. In the latter case, the reflected wave is amplified, especially, for ψ_{in} < π/4. For all the considered cases (for the “conventional” and MM biisotropic layer), the graphs in Figure 3 are weakly changed (changes of I_{r} do not exceed 1–3%) when varying chirality and nonreciprocity parameters in the ranges α, χ < 0.01.
Let us consider the dependences of the reflected wave polarization (characterized by ellipticity τ_{r} and polarization azimuth θ_{r}, Fig. 1) on the system parameters (Figs. 4 and 5). The numerical analysis shows that dependences τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) do not visually appear on the graphs for any combinations of the materials of layers 1, 3 (dielectric, ENZ, metallic) and incident TM or TE waves (τ_{in} = 0) for the nonchiral and reciprocal main layer (the data are not given in the graphs). In this case, ellipticity and polarization azimuth of the reflected wave are determined by the ones for the incident wave, and these quantities do not practically depend on the incidence angle. For the circularly polarized (RCP, LCP) incident wave and the absence of layers 1, 3, dependences τ_{r} (ψ_{in}) are neartolinear (Figs. 4(a,c,e)), solid curves). At the same time, for normal incidence (ψ_{in} = 0), the reflected wave has the circular polarization, which is opposite to one for the incident wave, τ_{r} = − τ_{in} = −1. For the case of oblique incidence, at ψ_{in} → π/2 the condition τ_{r} → τ_{in} is realized. When the spacer layers are absent (d_{1,3} = 0), quantity θ_{r} is decreased from value π/2 (obviously, that values θ_{r} = 0, π/2 are physically undistinguishable) down to π/4, Figure 4 (b,d,f). For the presence of layers 1, 3 and circular polarization of the incident wave (Fig. 4, all the curves besides the solid ones), dependences τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) can be nonlinear and characterized by the pronounced extrema. In this case, the significant effect on the polarization characteristics can be due to both double and single layers 1, 3, e.g. Figures 4 (c,d).
The numerical analysis also shows that the effect of layers 1, 3 on the reflected wave polarization is significantly weakened for the case . With that, dependences τ_{r} (ψ_{in}) are similar to the ones in Figure 4 for the case d_{1,3} = 0, and the condition θ_{r} ≈ − π/4 takes place for wide ranges of ψ_{in} (especially, for the dielectric or metallic spacer layers). When , the polarization characteristics under investigation are similar to the ones in Figure 4 for the case d_{1,3} = 0. As for the data considered in Figure 4, in this case, the effective conversion of the reflected wave polarization is realized.
When varying biisotropy parameters in wide ranges (α, χ < 0.01), including the cases of negative or and , the graphs in Figure 4, obtained for circular polarization of the incident radiation, have small changes (of the order of 1–2%). So, further, we consider the reflected wave polarization characteristics for the cases of the TM or TE incident wave (Fig. 5).
Data in Figure 5 illustrate the transformation of the reflected wave polarization (including the crosspolarization effects) for the chiral and reciprocal material of the main layer (for these data α = 5·10^{−3} (1 + i), χ = 0), various materials of the spacer layers and the TM or TE polarized incident wave. In these cases, the pronounced nonlinear and resonancelike dependences τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) can take place. To compare the polarization conversion regimes here and below, we consider the changes of quantities τ_{r,t} and θ_{r,t} (or θ_{r,t}) when varying the incidence angle and other parameters of the system (in the first turn, the spacer layers geometry). These changes also describe the differences of parameters τ_{r,t}, θ_{r,t} and the ones for the incident wave (τ_{in}, θ_{in}) in a number of cases below. The maximal changes of quantity τ_{r} (τ_{r} ≈ 0.6 ÷ 0.7) are realized for TM polarization in the case d_{1,3} = 0 (Fig. 5, the solid curves in panels (a,e,i)). For TE polarization, these values are τ_{r} ≈ 0.2 for dielectric layers 1, 3 and τ_{r} < 0.1 for ENZ and metallic layers 1, 3 in the cases d_{1,3} ≠ 0 (c,l) and d_{1} = 0, d_{3} ≠ 0 (g).
The maximal changes of quantity θ_{r} (θ_{r} ≈ 0.12π) take place under the condition d_{1,3} = 0 for the TM incident wave, panels (b,f,k) in Figure 5. For the TE polarized incident wave, we have θ_{r} ≈ 0.05π when d_{1,3} ≠ 0 for dielectric layers 1, 3 (d) and θ_{r} < 0.01π for ENZ and metallic layers 1, 3 in the cases d_{1,3} ≠ 0 (m) and d_{1} = 0, d_{3} ≠ 0 (h). For many cases in Figure 5, the double spacer layers, creating nanoresonator configurations (when d_{1,3} ≠ 0), determine (amplify) the transformation of the reflected wave polarization (especially, for the TE incident wave, panels (a–d,l,m)).
Some features of the data in Figure 5 take place under the additional variations of the parameters. For the chiral and nonreciprocal main layer material, a weakening of the considered polarization effects is realized for TM (TE) polarizations. Accounting for possible MM properties of the biisotropic layer, the following changes of the data in Figure 5 take place. In the case (), dependences τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) are practically absent for TM, TE polarizations of the incident wave and the considered materials of layers 1, 3. So, there is a peculiar suppression of the crosspolarization effects (for nonzero parameters α, χ), and this case is similar to the case of the nonchiral and reciprocal material of the main layer. When , , the corresponding graphs are similar to the ones in Figure 5 though the considered effects are slightly weakened (amplified) for the TM (TE) incident wave polarization.
Dependences τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) in Figure 5 are strongly pronounced for rather large incidence angles (ψ_{in} > 0.2π). The comparison of the data in Figures 5 and 3 also shows that the considered polarization effects mainly take place under the conditions of the approximate realization of Brewster's effect for the nanoresonator when the reflected wave intensity is small (I_{r} < 0.01 ÷ 0.02). According to the numerical analysis, with increasing values of α, χ (in compliance with the requirement of passivity of the main layer material [3,4]), the considered effects can be significantly amplified by choosing the spacer layers materials.
The reflection data (that correspond to the ones in Figs. 4, 5) for the case of the chiral EMNZ or chiral nihility material [26–28] of the main layer are given in Figure 6. The graphs in Figure 6 are obtained for metallic layers 1, 3 (similar results also take place for the ENZ or dielectric spacer layers). The graphs have features for small incidence angles. For ψ_{in} ≈ 0, functions I_{r} (ψ_{in}) have the pronounced minima for various polarizations, Figures 6 (a,d,g). With growing ψ_{in} values, these functions increase up to values I_{r} ≈ 0.9 and greater. For small ψ_{in}, the strong changes of the reflected wave polarization are realized, panels (b,c,e,f,h,i). At the same time, the significant (up to several tens of per cent) changes of τ_{r} and θ_{r} occur for a rather large reflection (I_{r} ≈ 0.1 ÷ 0.2 and greater, especially, for double or single layers 1, 3). In particular, the considerable rotation of the major axis of polarization ellipse for the TM or TE incident wave and the transition from the LCP to RCP (and vice versa) reflected wave are possible (panels (c,f) and (h), respectively).
The ψ_{in} ranges in Figure 6, when the strong polarization conversion takes place, correspond to the ones for the extreme dependences of birefringence and dichroism parameters of the field inside of the EMNZ biisotropic medium (see panels (d–i) in Figs. 3, 4 of [1]). So, according to the Figure 6 data, the significant modification of the reflected wave characteristics caused by the spacer layers takes place also in this case. The comparison of the data in Figures 4, 5 and 6 shows that in the latter case the values of τ_{r}, θ_{r}, I_{r} can significantly exceed (e.g. on the order and greater for the TM and TE polarizations) the ones for the case of the “conventional” biisotropic material of the main layer (Figs. 4 , 5).
According to the numerical analysis, the main layer material nonreciprocity (in the range of parameters χ < α) influences insignificantly I_{r} (ψ_{in}) graphs in Figure 6. However, τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) graphs for TM and TE polarizations can have significant changes in this case (quantities τ_{r}, θ_{r}, caused by nonreciprocity, can be of the order of 10–20% and greater, including cases of the corresponding increase of τ_{r}, θ_{r} values). Let us also note that the presence of dependences τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) in the Figures 5, 6 data (for TM and TE polarizations) is due to chirality and nonreciprocity of the main layer material. These dependences are absent in the case α = χ = 0 when the reflected wave polarization corresponds to the one for the incident TM or TE wave.
Fig. 3 The effect of the incidence angle on the reflected wave relative intensity for various exciting wave polarizations, geometries and materials of the spacer layers (here and below in similar figures, ψ_{in} is used as the abscissa axis for all the graphs). 
Fig. 4 Dependences of the reflected wave polarization parameters (ellipticity τ_{r} and polarization azimuth θ_{r}) on the incidence angle for RCP of the exciting wave (τ_{in} = 1), various geometries and materials of the spacer layers. 
Fig. 5 Dependences of the reflected wave polarization parameters on the incidence angle for TM and TE polarizations of the exciting wave (τ_{in} = 0), various geometries and materials of the spacer layers. χ = 0. 
Fig. 6 Dependences of the relative intensity and polarization parameters of the reflected wave on the incidence angle for the EMNZ biisotropic layer, various geometries of the metallic spacer layers, and incident wave polarizations. The values of parameters ε_{2} = 0.01 (1 + i), μ_{2} = 0.01 + 0.005i, α = 0.005 (1 + i), χ = 0 are used. 
5 Characteristic properties of transmitted radiation
The dependences of the transmitted wave relative intensity on the incidence angle for various exciting wave polarizations and spacer layers (that correspond to the Fig. 3 data for reflection) are given in Figure 7. The comparison of the graphs in Figures 3, 7 shows that the approximate relation I_{t} ≈ 1 − I_{r} takes place. So, the absorption of electromagnetic field energy inside of the multilayer is rather small. With that, a relative energy dissipation, Q ≈ 1 − I_{r} − I_{t}, does not exceed values 0.1 ÷ 0.2 for the cases in Figures 3, 7. Similar to the reflection case (Fig. 3), the spacer layers can significantly change dependences I_{t} (ψ_{in}) in Figure 7 (especially, when d_{1,3} ≠ 0). The transmission maxima, as a rule, correspond to Brewster's angles for reflection (panels (a–c), (g–i) in Figs. 3, 7). However, for the ENZ spacer layers, the approximate Brewster effect and I_{t} (ψ_{in}) maxima correspond to the slightly different incidence angles, panels (d,f) in Figures 3, 7. As for the reflected wave intensity (Fig. 3), the changes of I_{t} due to variation of parameters α, χ do not exceed 1–3% for the data in Figure 7.
Taking into account relation I_{t} ≈ 1 − I_{r}, the effects of possible MM properties of the biisotropic layer material mainly correspond to the reflection features considered in Figure 3. In particular, the Figure 7 graphs are weakly changed under the transition to the case . In the case , , dependences I_{t} (ψ_{in}) are very weak (I_{t} ≈ 10^{−4}) for all the data in Figure 7.
The dependences of the transmitted wave polarization (described by ellipticity τ_{t} and polarization azimuth θ_{t}, Fig. 1) on the incidence angle for various parameters are given in Figures 8, 9. As for the reflection case (Fig. 4), there are practically no dependences τ_{t} (ψ_{in}), θ_{t} (ψ_{in}) (τ_{t} ≈ τ_{in}, θ_{t} ≈ θ_{in} with high accuracy) for the case α = χ =0, TM or TE linearly polarized incident wave and various materials of layers 1, 3. For the circularly polarized incident wave and the absence of layers 1, 3, functions τ_{t} (ψ_{in}) are neartolinear and decreasing in the range τ_{t} = 1 ÷ 0.5 (the solid curves in panels (a,c,e) of Fig. 8). When ψ_{in} = 0, the transmitted wave polarization (RCP or LCP) corresponds to the one for the incident wave. The changes of θ_{t} are rather small for the case d_{1,3} = 0 (Figs. 8(b,d,f), the solid curves). The spacer layers provide the significant modification of the graphs of functions τ_{t} (ψ_{in}), θ_{t} (ψ_{in}), especially, when d_{1,3} ≠ 0 for dielectric and metallic layers 1, 3 (Figs. 8(a,e,b,f)) and for all the cases of ENZ layers (Figs. 8(c,d)).
Under the condition (the graphical data are not given) for all cases in Figure 8, besides the case d_{1,3} ≠ 0, the graphs have similar features. At the same time, when d_{1,3} ≠ 0, the considerable changes of the transmitted wave polarization take place for cases (a,b,e,f). As in the reflection case (Fig. 4), the variation of parameters α, χ in the ranges α, χ < 0.01 leads to the small changes of the graphs for all cases considered in Figure 8.
Figure 9 is a transmission “analogue” of Figure 5 (besides χ = 0, the same other parameters are used). Dependences τ_{t} (ψ_{in}), θ_{t} (ψ_{in}) for the data in Figure 9 are absent at α = χ = 0 when the transmitted wave polarization corresponds to the one for the incident TM or TE wave. According to Figure 9, the spacer layers can significantly amplify polarization conversion of the transmitted radiation, especially, for rather large incidence angles (ψ_{in} > π/4). The maximal changes of ellipticity τ_{t} (Δτ_{t} ≈ 1) and polarization azimuth (Δθ_{t} ≈ 0.5π) (in comparison with the ones for the incident wave, τ_{in}, θ_{in}) under the variation of ψ_{in} take place for the TM polarization and ENZ layers 1, 3 (the cases d_{1,3} ≠ 0 and d_{1} = 0, d_{3} ≠ 0 in panels (e,f) of Fig. 9). For the remaining cases in Figure 9, values Δτ_{t} ≈ 0.1, θ_{t} ≈ 0.02π and less are realized.
For the dielectric or metallic spacer layers, the corresponding graphs are qualitatively similar, panels (a–d) and (i–m), though the differences can take place for the separate cases, e.g. panels (c,d) and (l,m). As for the reflection case (Fig. 5), both double and single layers 1, 3 can amplify the polarization effects, e.g. panels (b,e,f,k,l) in Figure 9. The numerical analysis also shows that the main layer nonreciprocity (in the range of parameters χ < α) leads to the small changes of the Figure 9 graphs (the changes of τ_{t}, θ_{t} do not exceed 1–3%). The effects considered in Figure 9 also take place for the case (with small decreases of Δτ_{t}, Δθ_{t} values).
Figure 10 illustrates the transmitted wave properties (corresponding to the ones in Fig. 6) for the EMNZ biisotropic material of the main layer. It is seen, Figures 10(a,d,g), that functions I_{t} (ψ_{in}) have the pronounced maximum for small incidence angles (in particular, at ψ_{in} = 0 for the case d_{1,3} = 0). With growing ψ_{in}, these functions are monotonically decreasing. As for the reflection case (Fig. 6), there are strong dependences τ_{t} (ψ_{in}), θ_{t} (ψ_{in}) with one or several extrema for small ψ_{in}, Figures 10(b,c,e,f,h,i). With that, the considerable changes of polarization parameters depending on ψ_{in} take place (up to values Δτ_{t} ≈ 0.2 ÷ 0.4, Δθ_{t} ≈ 0.1π for TM or TE polarizations, panels (b,c,e,f), and Δτ_{t} ≈ 0.9, θ_{t} ≈ 0.2π for RCP (h,i)).
Comparing Figures 8–10, we see that quantities Δτ_{t}, θ_{t} are significantly greater (on the two orders and more for TM and TE polarizations) than the ones in Figures 8 and 9. These polarization transformations can correspond to the rather high intensities (I_{t} ≈ 0.5 and greater, especially, for the cases d_{1} ≠ 0 and (or) d_{3} ≠ 0). In particular, the RCP incident wave can be transformed into a nearlinearly polarized transmitted wave, Figures 10(h,i). As for the reflection case (Fig. 6), the regimes of the strong amplification of polarization effects (due to the spacer layers) are possible. The similar effects are also realized for the ENZ or dielectric spacer layers and for the case of chiral and reciprocal material of the main layer (χ = 0).
The dependences of the transmitted radiation parameters (corresponding to the Fig. 10 data) on the relative thickness of the main layer, d_{2}/λ, where λ is the incident radiation wavelength, are given in Figure 11. It is interesting that the graphs in Figures 10 and 11 are very similar. So, the increase of ψ_{in} is approximately equivalent to the growth of d_{2}/λ for the considered parameter ranges. According to the data in Figure 11, conditions 0 < ψ_{in} < 0.05π, d_{2}/λ < 0.5 are rather optimal for the polarization conversion. In doing so, values I_{t} > 0.1 ÷ 0.2 (panels (a,d,g) in Fig. 11), τ_{t} ≈ 0.6 ÷ 1, Δθ_{t} ≈ (0.1 ÷ 0.3) π (b,c,e,f,h,i) take place. In this case, the spacer layers can lead to the significant modification of the transmitted wave polarization, especially, for the cases d_{1,3} ≠ 0 and d_{1} = 0, d_{3} ≠ 0. Let us note that the EMNZ biisotropic main layer suppresses the multibeam interference effects (that are similar to the ones considered in [1]) for dependences I_{t} (d_{2}/λ).
Some changes of Figure 11 data under the parameter variations were also considered (the graphical data were not represented). For the case of the near to normal incidence (ψ_{in} ≈ 0), the dependences for quantities τ_{t}, θ_{t} in Figure 11 become near to linear and weakly decreasing (Δτ_{t} < 0.05, Δθ_{t} < 0.02π) for TM or TE incident wave, and condition τ_{t} ≈ τ_{in} takes place for the circular polarization. Similar features also take place for the “conventional” (nonEMNZ) biisotropic main layer (for ) under both the normal and oblique incidence.
Fig. 7 The effect of the incidence angle on the transmitted wave relative intensity for various polarizations of the exciting wave, geometries and materials of the spacer layers. 
Fig. 8 Dependences of the transmitted wave polarization parameters on the incidence angle for RCP of the exciting wave (τ_{in} = 1), various geometries and materials of the spacer layers. 
Fig. 9 Dependences of the transmitted wave polarization parameters on the incidence angle for TM and TE polarizations of the exciting wave (τ_{in} = 0), various geometries and materials of the spacer layers. χ = 0. 
Fig. 10 Dependences of the relative intensity and polarization parameters of the transmitted wave on the incidence angle for the EMNZ biisotropic main layer, various geometries of the metallic spacer layers, and incident wave polarizations. The parameters ε_{2} = 0.01 (1 + i), μ_{2} = 0.01 + 0.005i are used. 
Fig. 11 Dependences of the relative intensity and polarization parameters of the transmitted wave on the main layer relative thickness for the EMNZ biisotropic main layer, various geometries of the metallic spacer layers, and incident wave polarizations. The values of parameters correspond to Figure 10. ψ_{in} = 0.05π. 
6 Conclusion
The executed numerical modeling, based on the previous analytical results [1], generally confirms the idea to apply the considered subwavelength multilayer systems (as a rule, in the nanoresonator configurations), including the main biisotropic layer sandwiched between the spacer layers of various “conventional” or ENZ materials, for the control and amplification of chiroptical response in nanophotonics devices, effective polarization conversion. In particular, such nanoresonators can transform the incident electromagnetic wave with linear or circular polarization into the output waves with various polarization parameters.
The results of the numerical analysis have shown the main features of the reflection and transmission in the investigated systems. The energy and polarization characteristics of the reflected and transmitted waves can depend significantly on the electromagnetic properties and geometry of the spacer layers (the dielectric, ENZ, and metallic ones, with the thicknesses of the order of hundreds and tens of nanometers, respectively, Figs. 2, 3, 7). The dependences of energy (I_{r,t}) and polarization (τ_{r,t}, θ_{r,t}) characteristics of the radiation on the incidence angle (ψ_{in}) have been investigated in detail (Figs. 3–10). For the linear (TM or TE) polarizations of the exiting wave, the following main features take place: (i) the resonancelike dependences τ_{r} (ψ_{in}), θ_{r} (ψ_{in}) for the ψ_{in} ranges corresponding to the approximate realization of Brewster's effect (I_{r} < 0.01 ÷ 0.02, Fig. 5); (ii) the strong nonlinear dependences τ_{t} (ψ_{in}), θ_{t} (ψ_{in}), especially, for values ψ_{in} ≥ π/4 when I_{t} < 0.8 ÷ 0.9 (Fig. 9). Under the usually realizable condition ε_{2}, μ_{2} ≫ α, χ, the I_{r,t} changes due to the used values of chirality and nonreciprocity parameters are not exceed 1–3%. Parameters α, χ affect functions τ_{r,t} (ψ_{in}), θ_{r,t} (ψ_{in}) stronger for the linearly polarized exciting wave. As a rule, the chirality effects on the polarization characteristics (more pronounced for the transmitted radiation) are the most significant in comparison with the nonreciprocity effects.
The effects of various MM multilayer components on the reflection and transmission have been considered. In particular, in the transition from the “conventional” biisotropic layer () to the doublenegative MM one (), the output wave intensities (e.g. dependences I_{r,t} (ψ_{in})) are weakly changed. However, the polarizations and wave vectors of these waves can be much more “sensitive” to such a transition (see also [1]). So, the corresponding polarization effects can be used to identify the MM effective properties of chiral and biisotropic nanocomposites. With that, MM properties of the main layer material can both weaken (e.g. when ) and amplify the chiral and nonreciprocal effects.
Rather optimal conditions to transform polarization of the reflected or transmitted radiation (when quantities I_{r,t} and changes of τ_{r,t}, θ_{r,t} are maximal under variation of the system parameters and in comparison with the ones for the incident wave) are provided for the EMNZ biisotropic (in particular, EMNZ chiral or chiral nihility) material of the main layer. In these cases, the changes of polarization parameters, e.g. for the oblique incidence (ψ_{in} < 0.2π) of TM or TE polarized incident waves, are of the order of Δτ_{r} ≈ 0.7, Δθ_{r} ≈ 0.4π and Δτ_{t} ≈ 0.8, Δθ_{t} ≈ 0.2π for relative intensities I_{r} > 0.1 ÷ 0.2 and I_{t} > 0.5, respectively, and subwavelength thicknesses of the system (Figs 6, 10, 11). These Δτ_{t}, Δθ_{t} values can exceed significantly (on the one (two) order(s) and more for the reflection (transmission), correspondingly) the ones for the cases of the “conventional” (nonEMNZ) biisotropic or chiral main layer.
For the majority of the considered multilayer and incident wave parameters, the presence, variation of the materials and geometry of the spacer layers (especially, for the nanoresonator configurations) can provide the manifold opportunities to control and amplify the investigated (mainly, polarization) effects. Let us note that these effects can be considerably amplified for the thin biisotropic layer with the larger values of chirality and nonreciprocity parameters (that can be obtained using various approaches [29–34]) and applying modern optimization methods (e.g. [35,36]) to search desired parameters of the system. Furthermore, various approaches to reduce the size of chiral or biisotropic inclusions and nanoparticles in nanocomposites down to the size of separate molecules are actively developed (e.g. [6–9, 37–39]). The considered characteristic properties of polarization conversion in the biisotropic systems can take place for multiple devices on the basis of the investigated nanoresonators or their modifications. The proposed concept can be simply scaled for wide (not only optical) frequency ranges of coherent radiation and various types of the multilayer components. The obtained results (together with the ones in [1]) can be also relevant to the modeling and development of metasurfaces, polarization converters with reduced volumes of active material, ultrathin polarization modulators, chiral sensors, nanophotonics components of chiroptical spectroscopy and data processing systems.
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Cite this article as: Evgenii Starodubtsev, Reflection and transmission of nanoresonators including biisotropic and metamaterial layers: opportunities to control and amplify chiral and nonreciprocal effects for nanophotonics applications, EPJ Appl. Metamat. 10, 5 (2023)
All Figures
Fig. 1 The multilayer scheme and geometry of the corresponding electromagnetic boundary problem. The superstrate and substrate media are isotropic and semiinfinite. The multilayer includes spacer layers 1, 3 with thicknesses d_{1,3}, whose electromagnetic properties correspond to isotropic media (in particular, ENZ MMs), and the main biisotropic layer 2 with thickness d_{2}. Relative intensities I_{in}, I_{r}, I_{t}, ellipticities τ_{in}, τ_{r}, τ_{t}, polarization azimuths θ_{in}, θ_{r}, θ_{t}, and wave vectors k_{in}, k_{r}, k_{t} correspond to the incident, reflected, and transmitted waves, ψ_{in} and ψ_{t} are the angles of incidence and refraction, respectively. The fields inside of the multilayer are not shown. The inserts bounded by dotted lines illustrate orientations of the major axes of polarization ellipses of the incident, reflected and transmitted waves in the corresponding phase planes X'Y, X''Y, X'''Y, respectively. Additional notation is given in the main text. 

In the text 
Fig. 2 The effect of the real part of the spacer layers permittivity on the parameters of reflected (I_{r}, τ_{r}, θ_{r}) and transmitted (I_{t}, τ_{t}, θ_{t}) waves ( is used as the abscissa axis for all the graphs). The following values of parameters are used: d_{1,3} = 0.15 μm, d_{2} = 0.25 μm, , ε_{2} = 3 +0.01i, μ_{2} = 1.5 + 0.01i. The values of ψ_{in} and the incident wave polarization (RCP and LCP for right and left circular polarizations, here and below) are given above the corresponding graph panels. The other values of parameters correspond to the ones given in the main text (here and in the figures below). 

In the text 
Fig. 3 The effect of the incidence angle on the reflected wave relative intensity for various exciting wave polarizations, geometries and materials of the spacer layers (here and below in similar figures, ψ_{in} is used as the abscissa axis for all the graphs). 

In the text 
Fig. 4 Dependences of the reflected wave polarization parameters (ellipticity τ_{r} and polarization azimuth θ_{r}) on the incidence angle for RCP of the exciting wave (τ_{in} = 1), various geometries and materials of the spacer layers. 

In the text 
Fig. 5 Dependences of the reflected wave polarization parameters on the incidence angle for TM and TE polarizations of the exciting wave (τ_{in} = 0), various geometries and materials of the spacer layers. χ = 0. 

In the text 
Fig. 6 Dependences of the relative intensity and polarization parameters of the reflected wave on the incidence angle for the EMNZ biisotropic layer, various geometries of the metallic spacer layers, and incident wave polarizations. The values of parameters ε_{2} = 0.01 (1 + i), μ_{2} = 0.01 + 0.005i, α = 0.005 (1 + i), χ = 0 are used. 

In the text 
Fig. 7 The effect of the incidence angle on the transmitted wave relative intensity for various polarizations of the exciting wave, geometries and materials of the spacer layers. 

In the text 
Fig. 8 Dependences of the transmitted wave polarization parameters on the incidence angle for RCP of the exciting wave (τ_{in} = 1), various geometries and materials of the spacer layers. 

In the text 
Fig. 9 Dependences of the transmitted wave polarization parameters on the incidence angle for TM and TE polarizations of the exciting wave (τ_{in} = 0), various geometries and materials of the spacer layers. χ = 0. 

In the text 
Fig. 10 Dependences of the relative intensity and polarization parameters of the transmitted wave on the incidence angle for the EMNZ biisotropic main layer, various geometries of the metallic spacer layers, and incident wave polarizations. The parameters ε_{2} = 0.01 (1 + i), μ_{2} = 0.01 + 0.005i are used. 

In the text 
Fig. 11 Dependences of the relative intensity and polarization parameters of the transmitted wave on the main layer relative thickness for the EMNZ biisotropic main layer, various geometries of the metallic spacer layers, and incident wave polarizations. The values of parameters correspond to Figure 10. ψ_{in} = 0.05π. 

In the text 
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